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April  2017, 13(2): 825-834. doi: 10.3934/jimo.2016048

Robust design of sensor fusion problem in discrete time

Xiao Lan Zhu 1, , Zhi Guo Feng 2,, and  Jian Wen Peng 2,

1. 

LinJiang Middle School, KaiZhou District, Chongqing, China
2. 

College of Mathematical Sciences, Chongqing Normal University, Chongqing, China

* Corresponding author: Hong-Yong Fu

Received  July 2015 Revised  January 2016 Published  August 2016

In this paper, we consider a robust sensor scheduling problem which estimates the state of an uncertain process based on measurements obtained by a given set of noisy sensors, where the measurements of sensors are subject to external interference uncertainties. We formulate this problem into a minimax optimal control problem, which is equivalent to a semi-infinite programming problem with a dynamic system. A discretization method is used to solve this problem, where the computation is very large scale in general. We propose an approximation method to reduce the computational complexity. For illustration, two numerical examples are solved.

Keywords: Robust sensor fusion design, semi-infinite programming.
Mathematics Subject Classification: Primary: 49M37; Secondary: 90C34.
Citation: Xiao Lan Zhu, Zhi Guo Feng, Jian Wen Peng. Robust design of sensor fusion problem in discrete time. Journal of Industrial & Management Optimization, 2017, 13 (2) : 825-834. doi: 10.3934/jimo.2016048
References:
[1]

T. Abburi and S. Narasimhan, Optimal sensor scheduling in batch processes using convex relaxations and tchebycheff systems theory, IEEE Transactions on Automatic Control, 59 (2014), 2978-2983.  doi: 10.1109/TAC.2014.2351692.  Google Scholar

[2]

Z. L. Deng, Y. Gao, L. Mao, Y. Li and G. Hao, New approach to information fusion steady-state Kalman filtering, Automatica, 41 (2005), 1695-1707.  doi: 10.1016/j.automatica.2005.04.020.  Google Scholar

[3]

Z. G. Feng, K. L. Teo and Y. Zhao, Branch and bound method for sensor scheduling in discrete time, Journal of Industrial and Management Optimization, 1 (2005), 499-512.  doi: 10.3934/jimo.2005.1.499.  Google Scholar

[4]

Z. G. Feng, K. L. Teo, N. U. Ahmed, Y. Zhao and W. Y. Yan, Optimal fusion of sensor data for Kalman filtering, Discrete and ContinuousDynamical Systems, 14 (2006), 483-503.   Google Scholar

[5]

Z. G. Feng, K. L. Teo, N. U. Ahmed, Y. Zhao and W. Y. Yan, Optimal fusion of sensor data for discrete Kalman filtering, Dynamic Systems and Applications, 16 (2007), 393-406.   Google Scholar

[6]

Z. G. Feng, K. L. Teo and V. Rehbock, Hybrid method for a general optimal sensor scheduling problem in discrete time, Automatica, 44 (2008), 1295-1303.  doi: 10.1016/j.automatica.2007.09.024.  Google Scholar

[7]

F. N. Grigor'ev and N. A. Kuznetsov, Control of the observation process in continuous systems, Problems of Control and Information Theory, 6 (1977), 181-201.   Google Scholar

[8]

F. N. Grigor'ev, About the control of information processing in discrete automatic systems, Automation and Remote Control, 43 (1982).   Google Scholar

[9]

H. Kushner, On the optimum timing of observations for linear control systems with unknown initial state, IEEE Transactions on Automatic Control, 9 (1964), 144-150.   Google Scholar

[10]

H. W. J. Lee, K. L. Teo and L. S. Jennings, Control parametrization enhancing techniques for optimal discrete-valued control problems, Automatica, 35 (1999), 1401-1407.  doi: 10.1016/S0005-1098(99)00050-3.  Google Scholar

[11]

H. W. J. Lee, K. L. Teo and A. E. B. Lim, Sensor scheduling in continuous time, Automatica, 37 (2001), 2017-2023.  doi: 10.1016/S0005-1098(01)00159-5.  Google Scholar

[12]

Y. Li, L. W. Krakow, E. K. Chong and K. N. Groom, Approximate stochastic dynamic programming for sensor scheduling to track multiple targets, Digital Signal Processing, 19 (2009), 978-989.  doi: 10.1016/j.dsp.2007.05.004.  Google Scholar

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J. L. Liu, Y. Sun, J. Yang, W. Y. Liu and W. M. Chen, Optimal sensor scheduling for hybrid estimation, Journal of Central South University, 20 (2013), 2186-2194.  doi: 10.1007/s11771-013-1723-4.  Google Scholar

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B. M. Miller, Observation control for discrete-continuous stochastic systems, IEEE Transactions on Automatic Control, 45 (2000), 993-998.  doi: 10.1109/9.855571.  Google Scholar

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V. Malyavej, I. R. Manchester and A. V. Savkin, Precision missile guidance using radar/multiple-video sensor fusion via communication channels with bit-rate constraints, Automatica, 42 (2006), 763-769.  doi: 10.1016/j.automatica.2005.12.024.  Google Scholar

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A. S. Matveev and A. V. Savki, Optimal state estimation in networked systems with asynchronous communication channels and switched sensors, Journal of optimization theory and applications, 128 (2006), 139-165.  doi: 10.1007/s10957-005-7562-1.  Google Scholar

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A. V. Savkin, R. J. Evans and E. Skafidas, The problem of optimal robust sensor scheduling, Systems Control Lett., 43 (2001), 149-157.  doi: 10.1016/S0167-6911(01)00086-X.  Google Scholar

[18]

S. L. Sun, Distributed optimal component fusion weighted by scalars for fixed-lag Kalman smoother, Automatica, 41 (2005), 2153-2159.  doi: 10.1016/j.automatica.2005.06.014.  Google Scholar

[19]

K. L. Teo, C. Goh and K. Wong, A Unified Computational Approach to Optimal Control Problems, copublished in the United States with John Wiley & Sons, Inc., New York, 1991.  Google Scholar

[20]

C. Z. Wu, K. L. Teo and X. Y. Wang, Minimax optimal control of linear system with input-dependent uncertainty, Journal of the Franklin Institute, 351 (2014), 2742-2754.  doi: 10.1016/j.jfranklin.2014.01.012.  Google Scholar

[21]

C. Z. Wu, K. L. Teo and S. Y. Wu, Min-max optimal control of linear systems with uncertainty and terminal state constraints, Automatica, 49 (2013), 1809-1815.  doi: 10.1016/j.automatica.2013.02.052.  Google Scholar

[22]

M. Yavuz and D. Jeffcoat, An analysis and solution of the sensor scheduling problem, In Advances in Cooperative Control and Optimization, Springer Berlin Heidelberg, 369 (2007), 167–177. doi: 10.1007/978-3-540-74356-9_10.  Google Scholar

show all references

References:
[1]

T. Abburi and S. Narasimhan, Optimal sensor scheduling in batch processes using convex relaxations and tchebycheff systems theory, IEEE Transactions on Automatic Control, 59 (2014), 2978-2983.  doi: 10.1109/TAC.2014.2351692.  Google Scholar

[2]

Z. L. Deng, Y. Gao, L. Mao, Y. Li and G. Hao, New approach to information fusion steady-state Kalman filtering, Automatica, 41 (2005), 1695-1707.  doi: 10.1016/j.automatica.2005.04.020.  Google Scholar

[3]

Z. G. Feng, K. L. Teo and Y. Zhao, Branch and bound method for sensor scheduling in discrete time, Journal of Industrial and Management Optimization, 1 (2005), 499-512.  doi: 10.3934/jimo.2005.1.499.  Google Scholar

[4]

Z. G. Feng, K. L. Teo, N. U. Ahmed, Y. Zhao and W. Y. Yan, Optimal fusion of sensor data for Kalman filtering, Discrete and ContinuousDynamical Systems, 14 (2006), 483-503.   Google Scholar

[5]

Z. G. Feng, K. L. Teo, N. U. Ahmed, Y. Zhao and W. Y. Yan, Optimal fusion of sensor data for discrete Kalman filtering, Dynamic Systems and Applications, 16 (2007), 393-406.   Google Scholar

[6]

Z. G. Feng, K. L. Teo and V. Rehbock, Hybrid method for a general optimal sensor scheduling problem in discrete time, Automatica, 44 (2008), 1295-1303.  doi: 10.1016/j.automatica.2007.09.024.  Google Scholar

[7]

F. N. Grigor'ev and N. A. Kuznetsov, Control of the observation process in continuous systems, Problems of Control and Information Theory, 6 (1977), 181-201.   Google Scholar

[8]

F. N. Grigor'ev, About the control of information processing in discrete automatic systems, Automation and Remote Control, 43 (1982).   Google Scholar

[9]

H. Kushner, On the optimum timing of observations for linear control systems with unknown initial state, IEEE Transactions on Automatic Control, 9 (1964), 144-150.   Google Scholar

[10]

H. W. J. Lee, K. L. Teo and L. S. Jennings, Control parametrization enhancing techniques for optimal discrete-valued control problems, Automatica, 35 (1999), 1401-1407.  doi: 10.1016/S0005-1098(99)00050-3.  Google Scholar

[11]

H. W. J. Lee, K. L. Teo and A. E. B. Lim, Sensor scheduling in continuous time, Automatica, 37 (2001), 2017-2023.  doi: 10.1016/S0005-1098(01)00159-5.  Google Scholar

[12]

Y. Li, L. W. Krakow, E. K. Chong and K. N. Groom, Approximate stochastic dynamic programming for sensor scheduling to track multiple targets, Digital Signal Processing, 19 (2009), 978-989.  doi: 10.1016/j.dsp.2007.05.004.  Google Scholar

[13]

J. L. Liu, Y. Sun, J. Yang, W. Y. Liu and W. M. Chen, Optimal sensor scheduling for hybrid estimation, Journal of Central South University, 20 (2013), 2186-2194.  doi: 10.1007/s11771-013-1723-4.  Google Scholar

[14]

B. M. Miller, Observation control for discrete-continuous stochastic systems, IEEE Transactions on Automatic Control, 45 (2000), 993-998.  doi: 10.1109/9.855571.  Google Scholar

[15]

V. Malyavej, I. R. Manchester and A. V. Savkin, Precision missile guidance using radar/multiple-video sensor fusion via communication channels with bit-rate constraints, Automatica, 42 (2006), 763-769.  doi: 10.1016/j.automatica.2005.12.024.  Google Scholar

[16]

A. S. Matveev and A. V. Savki, Optimal state estimation in networked systems with asynchronous communication channels and switched sensors, Journal of optimization theory and applications, 128 (2006), 139-165.  doi: 10.1007/s10957-005-7562-1.  Google Scholar

[17]

A. V. Savkin, R. J. Evans and E. Skafidas, The problem of optimal robust sensor scheduling, Systems Control Lett., 43 (2001), 149-157.  doi: 10.1016/S0167-6911(01)00086-X.  Google Scholar

[18]

S. L. Sun, Distributed optimal component fusion weighted by scalars for fixed-lag Kalman smoother, Automatica, 41 (2005), 2153-2159.  doi: 10.1016/j.automatica.2005.06.014.  Google Scholar

[19]

K. L. Teo, C. Goh and K. Wong, A Unified Computational Approach to Optimal Control Problems, copublished in the United States with John Wiley & Sons, Inc., New York, 1991.  Google Scholar

[20]

C. Z. Wu, K. L. Teo and X. Y. Wang, Minimax optimal control of linear system with input-dependent uncertainty, Journal of the Franklin Institute, 351 (2014), 2742-2754.  doi: 10.1016/j.jfranklin.2014.01.012.  Google Scholar

[21]

C. Z. Wu, K. L. Teo and S. Y. Wu, Min-max optimal control of linear systems with uncertainty and terminal state constraints, Automatica, 49 (2013), 1809-1815.  doi: 10.1016/j.automatica.2013.02.052.  Google Scholar

[22]

M. Yavuz and D. Jeffcoat, An analysis and solution of the sensor scheduling problem, In Advances in Cooperative Control and Optimization, Springer Berlin Heidelberg, 369 (2007), 167–177. doi: 10.1007/978-3-540-74356-9_10.  Google Scholar

Figure 1.  Optimal solutions in the first example
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Figure 2.  he cost function values in the first example
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Figure 3.  Optimal solutions of the second example
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